Unit 4 – Number Patterns and Fractions
Lesson 4 – Ordering Mixed Numbers Improper Fractions
Black – Ordering Mixed Numbers Improper Fractions
Once you feel comfortable with today’s lesson topic, the following problems can help
you get better at confronting problems that involve fractions. Give them a shot if
you’re up to the challenge!
1. Jumping Rope
My jumping rope was cut in half, half was thrown away. The other
half was cut again one third along the way. The longer part (ten feet
long) is what I use to play with.
How long was my jumping rope when I began today?
This problem was created by students at Western Oregon
University in the spring of 2002
2. Wheels R Us
One fourth of the vehicles at Danielle's Cycle Shop are tricycles. The rest are
bicycles. Danielle counted a total of 45 wheels in her shop.
How many bicycles does she have?
How many tricycles does she have?
Explain how you found your answer. Show how you know you are correct.
Extra: Danielle didn't count the pairs of training wheels on the back of 1/5 of the
bikes. How many total wheels are there if training wheels are included?
Tell how you figured it out.
3. How Old is Mia?
One day in math class Mia invented a problem for her classmates to solve. She said,
“My mom’s and dad’s ages are 3 years apart. I’m exactly 1/4 of my mom’s age, and I’m
24 years younger than Dad. How old am I?”
Mia was surprised when her classmates came up with two different ages that fit her
clues.
What answers did her classmates find?
Explain how you solved the problem. Show how you know both answers are correct.
Extra: Suppose that Mia is 1/3 of her mother's age. Find two other numbers to
replace the differences of 3 and 24 that still result in two possible ages for Mia.
4. Filling Beauty's Seats
Tickets to Fairview Elementary School's production of Beauty and the Beast went on
sale this week. The school theater has 24 rows of 16 seats each.
1/3 of all the seats have been sold to students for $3 each.
• 1/4 of them have been sold to adults for $5 each.
• 1/6 of them were given to the teachers.
•
1
Unit 4 – Number Patterns and Fractions
Lesson 4 – Ordering Mixed Numbers Improper Fractions
1. If everyone who already has a ticket goes to the show, what fraction of
the seats in the theater will be filled?
2. How many seats are still available?
Extra: How much money has been collected so far? If all the remaining seats are sold
to students, how much money will be raised altogether?
5. A Pound of Tea
Jonathan and Hazel love to drink tea. When they are both at home, a pound of tea
lasts two weeks. Occasionally Hazel visits her mother in Australia. It takes six weeks
for Jonathan alone to use up a pound of tea.
How long should a pound of tea last Hazel if she were alone?
Remember to explain how you solved the problem.
Extra: In 2006 Hazel visited her mother for six weeks in the spring. In the summer
Jonathan gave up drinking tea for six weeks. Their favorite tea costs $9.00 per
pound. About how much did the couple spend on tea during the year? Explain.
Solutions
1. The original length of the jumping rope was 30 feet.
After reading the riddle I decided to start doing the problem backwards. The riddle
said that they cut the rope one third down the length after cutting it in half and
throwing half away. Since the final piece was 10 feet long I just needed to add half
of that.
---------- + -------------------- = -----------------------------5 feet
10 feet
15 feet
Then they said it was cut in half, so all I needed to do was multiply 15 x 2 to equal 30
feet. Which was the original length.
2. We are told ¼ of the vehicles are tricycles. So ¾ of the vehicles must be bicycles.
This means there are 3 bicycles (or 6 wheels) for each tricycle (3 wheels). For every
tricycle removed from the total vehicles (45 wheels) we must also take away 3
bicycles. We can do this 5 times before all vehicles (45 wheels) are taken away. So
Danielle has 5 tricycles and 15 bicycles.
5 tricycles x 3 wheels per tricycle = 15 wheels
15 bicycles x 2 wheels per bicycle = 30 wheels
5 tricycles + 15 bicycles = 20 vehicles total
15 wheels + 30 wheels = 45 wheels total
2
Unit 4 – Number Patterns and Fractions
Lesson 4 – Ordering Mixed Numbers Improper Fractions
This is correct because 5 tricycles out of 20 vehicles equal ¼. And 3 bicycles out of
20 vehicles equal ¾.
5 tricycles / 20 vehicles = 1 tricycle / 4 vehicles
15 bicycles / 20 vehicles = 3 bicycles / 4 vehicles
Extra:
Danielle has 15 bicycles. 1/5 of 15 bicycles equal 3 bicycles. 3 bicycles with training
wheels equal 6 additional wheels. 6 training wheels + 45 wheels = 51 wheels total.
3. I started with a guess that Mia was 7 years old and used guess and check to see if
my answer was correct. I started with 7 years old because I started word problems
in second grade and that was when I was about 7 years old. I knew moms can be
either older or younger than dads, so I knew I needed to add or subtract 3 to get
moms age.
I knew that Dad = Mia + 24.
7 + 24 = 31, so 31 = dad.
I knew that the difference between Dad and Mom’s age was 3 years:
31 – 3 = 28, so Mom is 28.
I knew that Mom / 4 = Mia.
28/4=7, so Mia is 7. So 7 is a correct age.
Next I tried 8 but I quickly realized that using an even number for Mia’s age made
Mom’s age an odd number and odd numbers are not divisible by 4. 8 + 24 = 32 + 3 =
35, so this would be mom’s age and it is not divisible by 4.
I tried 9 because it is the next highest odd number.
9 + 24 = 33, so 33 = dad
33 + 3 = 36, so Mom is 36
36/4 = 9, so Mia is 9. So 9 is a correct age.
4. The question is what fraction of the total number of seats in the theater are filled
and how many seats are left. I know that there are 384 seats. 24 rows X 16 seats
per row = 384 seats total. I also know that not all the seats filled.
1/3 + 1/4 + 1/6 = 4/12 + 3/12 + 2/12 = 9/12 of the total number of seats.
First I divided 384 by 3 to get the number of seats for kids.
384/ 3 = 128 seats
Then I divided 384 by 4 to get the number of seats for adults.
384/ 4 = 96 seats
I also divided 384 by 6 to get the number of seats for teachers.
384 / 6 = 64 seats
3
Unit 4 – Number Patterns and Fractions
Lesson 4 – Ordering Mixed Numbers Improper Fractions
The total number of seats equals 288.
128 + 96 + 64 = 288.
The fractional representation of seats filled to total seats is: 288/384 which
reduces to 72/96 to 6/8 to 3/4 of the total.
If I subtract 288 from 384 I get 96 seats left if everyone goes to the show.
Extra: I multiplied each seat number by the cost to the seat holder and got a total
of $864. 128 kid seats x $3 = $384 and 96 adult seats x $5 = $480. The teacher's
tickets were free.
The 96 seats if they were sold to students they would earn another $288. 96 seats
x $3 = $288. This added to the original amount would earn them $1152 or $864 +
$288 = 1152.
5. I knew that when Jonathan and Hazel were at home it took them 2 weeks to finish
drinking a pound of tea and when Jonathan was alone, it took him 6 weeks to finish
drinking a pound of tea. Using this information I had to find the time it took Hazel to
drink a pound of tea.
I knew that Jonathan drank a pound of tea in 6 weeks so in 1 week he must have
drank 1/6 lb. of tea and in 2 weeks he should have drank 2/6 lb. of tea or simplified
to 1/3 lb. of tea. 2 weeks was the time it took them both to finish a pound of tea so,
the amount of tea Hazel drank in 2 weeks must equal 2/3 since that is the
difference of the amount of tea Jonathan drank(1/3 lb.) from the total amount of
tea(1 lb.). Now I had to find the time for Hazel herself to finish a pound of tea. In 1
week Hazel drank 1/3 lb., in 2 weeks Hazel drank 2/3 lb., so in 3 weeks she must have
drank 3/3 lb which is equal to 1 pound. Therefore, it took 3 weeks for Hazel to drink
1 pound of tea.
EXTRA: I knew that there are about 52 weeks in a year. Hazel did not drink tea for
6 weeks and neither did Jonathan. I had to change those 6 weeks to 3 weeks because
when Jonathan did not drink, Hazel drank and when Hazel did not drink, Jonathan
drank. That equals a total of 6 weeks without tea in that year. To find out how many
weeks they drank tea in the year you have to subtract 6 (the number of weeks they
did not drink tea) from 52 (the number of weeks in a year which would equal 46
weeks. This had to be divided by 2 because it took 2 weeks for them to finish
drinking a pound of tea which would equal 23 weeks. Their favorite tea costs $9.00
and to find how much they spent on tea, you have to multiply 23(the number of weeks
they drank tea) x 9(the cost of a pound of tea).
23 x 9 = 207
The answer was 207, so the amount of money they spent that year on tea was $207.
4
Unit 4 – Number Patterns and Fractions
Lesson 4 – Ordering Mixed Numbers Improper Fractions
Bibliography Information
Teachers attempted to cite the sources for the problems included in this problem set. In some cases,
sources may not have been known.
Problems
Bibliography Information
1-5
The Math Forum @ Drexel
(http://mathforum.org/)
5